TWISTED CONJUGACY IN LINEAR ALGEBRAIC GROUPS
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Springer Science+Business Media New York (2020)
TWISTED CONJUGACY IN LINEAR ALGEBRAIC GROUPS A. BOSE∗
S. BHUNIA
Department of Mathematics Department of Mathematics Indian Institute of Science Indian Institute of Science Education and Research (IISER) Education and Research (IISER) Mohali, Knowledge City, Sector 81 Mohali, Knowledge City, Sector 81 S.A.S. Nagar 140306, Punjab, India S.A.S. Nagar 140306, Punjab, India [email protected]
[email protected]
Abstract. Let k be an algebraically closed field, G a linear algebraic group over k and ϕ ∈ Aut(G), the group of all algebraic group automorphisms of G. Two elements x, y of G are said to be ϕ-twisted conjugate if y = gxϕ(g)−1 for some g ∈ G. In this paper we prove that for a connected non-solvable linear algebraic group G over k, the number of its ϕ-twisted conjugacy classes is infinite for every ϕ ∈ Aut(G).
Introduction Let G be a group and ϕ an endomorphism of G. Two elements x, y ∈ G are said to be ϕ-twisted conjugate, denoted by x ∼ϕ y, if y = gxϕ(g)−1 for some g ∈ G. Clearly, ∼ϕ is an equivalence relation on G. The equivalence classes with respect to this relation are called ϕ-twisted conjugacy classes or Reidemeister classes of ϕ. If ϕ = Id, then the ϕ-twisted conjugacy classes are the usual conjugacy classes. Let [x]ϕ denote the ϕ-twisted conjugacy class containing x ∈ G and R(ϕ) := {[x]ϕ | x ∈ G}. The cardinality of R(ϕ), denoted by R(ϕ), is called the Reidemeister number of ϕ. A group G is said to have the R∞ -property if R(ϕ) is infinite for every automorphism ϕ of G. The problem of determining groups which have the R∞ -property is an active area of research begun by Fel’shtyn and Hill [3] although the study of twisted conjugacy can be traced back to the works of Gantmacher in [6]. The reader may refer to [5] and the references therein for more literature. The R∞ -property of irreducible lattices in a connected semisimple Lie group of real rank at least 2 has been studied by Mubeena and Sankaran (see [11, Thm. 1]). Nasybullov showed that if K is an integral domain of zero characteristic and Aut(K) is torsion then GLn (K) and SLn (K) (for n > 2) have the R∞ -property (see [12]). A Chevalley group G (resp. a twisted Chevalley group G0 ) over a field K of characteristic zero possesses the R∞ -property if the transcendence degree of K over Q is finite, DOI: 10.1007/S00031-020-09626-9 Bose is supported by DST-INSPIRE Faculty fellowship (IFA DST/INSPIRE/04/ 2016/001846). Received May 4, 2020. Accepted September 14, 2020. Corresponding Author: A. Bose, e-mail: [email protected] ∗
S. BHUNIA, A. BOSE
see [5, Thm. 3.2] (resp. [1, Thm. 1.2]). It is worth mentioning that a reductive linear algebraic group G over an algebraically closed field k of characteristic zero possesses the R∞ -property if the transcendence degree of k over Q is finite and the radical of G is a proper subgroup of G (see [5, Thm. 4.1]). The converse of the latter holds if the group G is a Chevalley group of classical type (An , Bn , Cn , Dn ) as shown in [14, Thm. 8], wh
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